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68 Vector Basics
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That s the same as a b Let s call this Cartesian difference vector d = (xd,yd) Then xd = 0 and yd = 2 21/2 Using the Cartesian-to-polar conversion table, we can see that qd = p /2 and rd = (xd2 + yd2)1/2 = [02 + (2 21/2)2]1/2 = 2 21/2 The polar ordered pair is therefore a b = [(p /2),(2 21/2)] The first coordinate is the angle in radians The second coordinate is the magnitude in linear units
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By now you might wonder, What s the difference between a polar vector sum and a Cartesian vector sum Or a polar vector negative and a Cartesian vector negative Or a polar vector difference and a Cartesian vector difference If we start with the same vector or vectors, shouldn t we get the same vector when we re finished calculating, whether we do it the polar way or the Cartesian way That s an excellent question The answer is yes The mathematical methods differ, but the resultant vectors are equivalent whether we work them out the polar way or the Cartesian way
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Draw polar coordinate diagrams of the vector addition and subtraction facts we worked out in this section
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The original two polar vectors were a = (qa,ra) = (p /4,2) and b = (qb,rb) = (7p /4,2)
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The Polar Way
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We found their polar sum to be a + b = [0,(2 21/2)] and their polar difference to be a b = [(p /2),(2 21/2)] When we converted the two vectors to Cartesian form, we got a = (21/2,21/2) and b = (21/2, 21/2) We found their Cartesian sum to be a + b = [(2 21/2),0] and their Cartesian difference to be a b = [0,(2 21/2)] We can illustrate the original vectors, the vector sum, the negative of the second vector, and the vector difference in four diagrams:
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Figure 4-7 shows the polar sum, including a, b, and a + b
Figure 4-7 Polar sum of two vectors Each radial
division represents 1/2 unit
70 Vector Basics
Figure 4-8 shows the polar difference, including a, b, b, and a b
p /2
a b = a + ( b) = [p /2, (2 21/2)] -b a = (p /4, 2)
b = (7p /4, 2) This vector is converted to Cartesian form in the subtraction process
3p /2
Figure 4-8 Polar difference between two vectors Each radial
division represents 1/2 unit
Figure 4-9 shows the Cartesian sum, including a, b, and a + b
a = (21/2, 21/2)
a + b = [(2 21/2), 0] x
Each axis division is 1/2 unit
b = (21/2, -21/2)
Figure 4-9 Cartesian sum of two vectors Each axis
division represents 1/2 unit
Practice Exercises
Figure 4-10 shows the Cartesian difference, including a, b, b, and a b
a b = a + ( b) = [0, (2 21/2)] b = ( 21/2, 21/2) a = (21/2, 21/2)
Each axis division is 1/2 unit
b = (21/2, 21/2)
Figure 4-10
Cartesian difference between two vectors Each axis division represents 1/2 unit
Practice Exercises
This is an open-book quiz You may (and should) refer to the text as you solve these problems Don t hurry! You ll find worked-out answers in App A The solutions in the appendix may not represent the only way a problem can be figured out If you think you can solve a particular problem in a quicker or better way than you see there, by all means try it! 1 Consider two vectors a and b in the Cartesian plane, with coordinates defined as follows: a = ( 3,6) and b = (2,5) Work out, in strict detail, the Cartesian vector sums a + b, b + a, a b, and b a 2 A vector is defined as the zero vector (denoted by a bold numeral 0) if and only if its magnitude is equal to 0 In the Cartesian plane, the zero vector is expressed as the ordered pair (0,0) Show that when a vector is added to its Cartesian negative in either order, the result is the zero vector
72 Vector Basics
3 Imagine two arbitrary vectors a and b in the Cartesian plane, with coordinates defined as follows: a = (xa,ya) and b = (xb,yb) Show that the vector b a is the Cartesian negative of the vector a b 4 Find the Cartesian sum of the vectors a = (4,5) and b = ( 2, 3) Compare this with the sum of their negatives a = ( 4, 5) and b = (2,3) 5 Prove that Cartesian vector negation distributes through Cartesian vector addition That is, show that for two Cartesian vectors a and b, it s always true that (a + b) = a + ( b) 6 Find the polar sum of the vectors a = (p /2,4) and b = (p,3) 7 Find the polar negative of the vector a + b from the solution to Problem 6 8 Find the polar negatives a and b of the vectors stated in Problem 6 9 Find the polar sum of the vectors a and b from the solution to Problem 8 Compare this with the solution to Problem 7 10 Find the polar differences a b and b a between the vectors stated in Problem 6